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Properties

Label 2-950-95.88-c1-0-26
Degree $2$
Conductor $950$
Sign $-0.898 + 0.439i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (0.239 + 0.0641i)3-s + (−0.866 − 0.499i)4-s + (0.123 − 0.214i)6-s + (2.17 − 2.17i)7-s + (−0.707 + 0.707i)8-s + (−2.54 − 1.46i)9-s − 0.518·11-s + (−0.175 − 0.175i)12-s + (−1.11 − 4.17i)13-s + (−1.53 − 2.66i)14-s + (0.500 + 0.866i)16-s + (1.36 + 0.366i)17-s + (−2.07 + 2.07i)18-s + (−3.08 + 3.08i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.138 + 0.0370i)3-s + (−0.433 − 0.249i)4-s + (0.0505 − 0.0875i)6-s + (0.822 − 0.822i)7-s + (−0.249 + 0.249i)8-s + (−0.848 − 0.489i)9-s − 0.156·11-s + (−0.0505 − 0.0505i)12-s + (−0.310 − 1.15i)13-s + (−0.411 − 0.712i)14-s + (0.125 + 0.216i)16-s + (0.332 + 0.0889i)17-s + (−0.489 + 0.489i)18-s + (−0.707 + 0.706i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.898 + 0.439i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.898 + 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299231 - 1.29224i\)
\(L(\frac12)\) \(\approx\) \(0.299231 - 1.29224i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
19 \( 1 + (3.08 - 3.08i)T \)
good3 \( 1 + (-0.239 - 0.0641i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.17 + 2.17i)T - 7iT^{2} \)
11 \( 1 + 0.518T + 11T^{2} \)
13 \( 1 + (1.11 + 4.17i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.36 - 0.366i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (1.28 - 0.343i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.52 + 6.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.22iT - 31T^{2} \)
37 \( 1 + (7.09 + 7.09i)T + 37iT^{2} \)
41 \( 1 + (2.55 - 1.47i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.18 + 8.14i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.70 + 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.49 - 9.32i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.414 + 0.718i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.88 + 5.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.44 + 0.922i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.71 - 5.60i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.501 - 1.87i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.10 + 5.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.80 - 5.80i)T + 83iT^{2} \)
89 \( 1 + (2.05 - 3.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.86 - 6.97i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06446493170979850184297124102, −8.761338207858139952978113114270, −8.189997022093653812891824301464, −7.31793443712718608018141403718, −5.98623586780847890490369485969, −5.22459940371700293588477631116, −4.12063521014953093797120255855, −3.28396086558051783729259180262, −2.06900146642158585158412471146, −0.55741255345291502403755145803, 1.93945204614561730488016308350, 3.04467965589010788372672887577, 4.57917197527689353986789903279, 5.10937434267826887663300790778, 6.10489020716753495078206625234, 6.97738115879959010802899464386, 8.004211421809085219166353726010, 8.616162473737861096951671016466, 9.193957613446358797622937209650, 10.36653446357411481503250132807

Graph of the $Z$-function along the critical line